Most extant localization theories for spaces, spectra and diagrams
of such can be derived from a simple list of axioms which are verified
in broad generality. Several new theories are introduced, including
localizations for simplicial presheaves and presheaves of spectra at
homology theories represented by presheaves of spectra, and a theory
of localization along a geometric topos morphism. The
$f$-localization concept has an analog for simplicial presheaves, and
specializes to the $\hbox{\Bbbvii A}^1$-local theory of
Morel-Voevodsky. This theory answers a question of Soul\'e concerning
integral homology localizations for diagrams of spaces.